阅读说明：本技术 一种对称的分数阶变压器构造电路及构造方法 (Symmetrical fractional order transformer construction circuit and construction method ) 是由 谢帆 杨晨 张波 陈艳峰 丘东元 肖文勋 黄子田 于 2021-01-25 设计创作，主要内容包括：本发明公开了一种对称的分数阶变压器构造电路及构造方法,包括一次侧回路和二次侧回路。一次侧回路包括n个原边电阻和n个变压器的原边绕组,所述的第i个原边电阻与第i个变压器的原边绕组串联形成第i条原边支路,一次侧的n条原边支路并联后形成A、B端子；二次侧回路包括n个副边电阻和n个变压器的副边绕组,所述的第i个副边电阻与第i个变压器的副边绕组串联形成第i条副边支路,二次侧的n条副边支路并联后形成C、D端子。本发明能够真实的模拟一个对称的分数阶变压器,实现了对称的分数阶变压器的构造；将变压器元件扩展到分数阶领域,对电路理论与实际研究具有重要意义。(The invention discloses a symmetrical fractional order transformer construction circuit and a construction method thereof. The primary side loop comprises n primary side resistors and n primary side windings of transformers, the ith primary side resistor and the ith primary side winding of the transformer are connected in series to form an ith primary side branch, and the n primary side branches of the primary side are connected in parallel to form an A, B terminal; the secondary side loop comprises n secondary side resistors and n secondary side windings of transformers, the ith secondary side resistor and the secondary side winding of the ith transformer are connected in series to form an ith secondary side branch, and the n secondary side branches on the secondary side are connected in parallel to form an C, D terminal. The invention can truly simulate a symmetrical fractional order transformer, and realizes the construction of the symmetrical fractional order transformer; the transformer element is expanded to the fractional order field, and the method has important significance for circuit theory and actual research.)

1. A symmetrical fractional order transformer structure circuit is characterized by comprising a primary side loop and a secondary side loop;

the primary side loop comprises n primary side resistors R_pAnd n primary windings of transformer T, i primary resistance R_piThe voltage regulator is connected with a primary winding of an ith transformer in series to form an ith primary side branch, and n primary side branches are connected in parallel to form A, B two terminals;

the secondary side loop comprises n secondary side resistors R_sAnd secondary windings of the n transformers T, an ith secondary resistor R_siThe voltage regulator is connected with a secondary winding of the ith transformer in series to form an ith secondary branch, and the n secondary branches are connected in parallel to form C, D two terminals.

2. The circuit according to claim 1, wherein the n transformers T each have an air gap and have the same turns ratio.

3. The symmetrical fractional order transformer configuration circuit of claim 2, wherein the primary resistance R is varied_pSecondary side resistance R_sAnd parameters of the transformer T to realize symmetrical division of different ordersSeveral orders of transformer characteristics.

4. A method of constructing a symmetrical fractional order transformer construction circuit according to claim 3, comprising the steps of:

s1, determining basic parameters of a symmetrical fractional order transformer corresponding to a construction circuit;

s2, determining the turn ratio of a transformer T in the construction circuit;

s3, determining the number n of primary side resistors and the number R of the primary side resistors in the constructed circuit_piSecondary side resistance R_siAnd the ith transformer T_iExcitation inductance L_miA value of (d);

s4, determining primary side leakage inductance L of n transformers T in structural circuit_pSecondary leakage inductance L_sThe values of the primary winding resistance and the secondary winding resistance.

5. The method as claimed in claim 4, wherein the basic parameters include the order α of the transformer and the inductance L of the primary side inductor₁Inductance value L of secondary side inductor₂；

According to inductance L of primary side inductance₁Inductance value L of secondary side inductor₂Determining the turn number transformation ratio k of n transformers T in the constructed circuit to be:

6. the method of claim 5, wherein the order N of the filter is selected and the frequency bin (ω) is selected based on Oustaloup rational approximation_b,ω_h) Inductance to primary side inductance of L₁And the fractional order inductance element with the order of alpha is reasonably approximated to obtain an approximate equivalent circuit of the corresponding fractional order inductance, wherein omega is_bIs the lower limit of said frequency band, ω_hIs that it isAn upper frequency band limit of (d);

7. the method as claimed in claim 6, wherein the approximately equivalent circuit of fractional order inductor comprises N resistors R connected in series_iAnd an inductance L_iThe branches are connected in parallel.

8. The method of claim 7, wherein the number N of primary resistors is the same as the filter order N.

9. The method of claim 8, wherein the fractional inductance is approximately equivalent to the equivalent circuit resistance R_iFor constructing a primary resistor R in the circuit_pi(ii) a Inductance L of approximate equivalent circuit of fractional order inductance_iFor constructing the ith transformer T in the circuit_iExcitation inductance L_mi(ii) a Secondary side resistance R_siIs a primary side resistor R_piIs/are as followsAnd (4) doubling.

10. The method of claim 9, wherein the primary side leakage inductance L is_pSet as excitation inductance L_m1% -3% of (A), secondary leakage inductance L_sSet as primary side leakage inductanceThe primary winding resistance and the secondary winding resistance are both set to be less than 10m omega.

Technical Field

The invention relates to the technical field of fractional order device construction, in particular to a symmetrical fractional order transformer construction circuit and a construction method.

Background

In the fields of science and engineering, recent research shows that the use of fractional calculus to describe many materials, processes and natural phenomena in nature is simpler and more accurate than integer calculus, and furthermore, the introduction of fractional calculus can also achieve effects and performances that cannot be achieved by integer calculus, for example, enabling circuits to have higher degrees of freedom and flexibility (Jiang y.w., Zhang b., Shu, x.j., et al.sectional-order auto-dynamics circuits with order of der lar. adv.res, vol.25, pp.217-225,2020.).

In the field of electrical engineering, elements such as inductors, capacitors, transformers, etc. are generally considered to be of integer order, however studies have shown that these elements are all fractional in nature (westerund s., Ekstam l.capacitor theory. ieee trans. diector.electric. instrument. fault, vol.1, pp.826-839,1994. ", reference 3" westerund s.dead materials Memory | physics. scr, vol.43, pp.174-179,1991.), and since there is no single fractional element currently commercially available, obtaining an approximately ideal fractional element by a reasonable construction method is an effective solution for research purposes.

At present, a common construction method of a fractional inductance includes a construction method of realizing a fractional inductance of 0-1 order based on an RL chain impedance method and using a transconductance operational amplifier, and a construction method of realizing a fractional inductance of 1-2 orders based on an impedance conversion circuit (GIC) (tsirikou g., pseudosolinos. c, Freeborn t.j., et al.emulation of current fractional-order capacitors and impedance using OTA polarizations.microelectron.j., vol.55, pp.70-81,2016. ", reference 5" tripath m.c., nodal d., biswis k., et al.experimental reactions and analysis of fractional-order inductors-inductors, and plug. 1196,2015.3.v.). Fractional Order capacitors are similarly constructed, including RC-chain based fractional reactance and transconductance based operational amplifiers (Sierocituk D., Podlubny I., Petras I. Experimental evaluation of Variable-Order behavor of amplifiers and New amplifiers IEEE Transactions on Control systems technology, vol.21, pp.459-466,2013. ", reference 7" Tsiremokou G., Psychalinos.C., Elwakil A.S. Emulation of a constant phase electron using operational amplifiers, Analog integer Circ processors, 85.413-423,2015.). However, for the fractional order transformer element, the magnetic energy coupling and the energy conversion are involved, so that the structural research of the fractional order transformer is relatively deficient, and the fractional order transformer element is still more at the level of modeling and characteristic analysis of the fractional order transformer.

Disclosure of Invention

The invention aims to overcome the defects of the prior art and provides a symmetrical fractional order transformer structure circuit.

The invention is realized by at least one of the following technical schemes.

A symmetrical fractional order transformer structure circuit comprises a primary side loop and a secondary side loop;

the primary side loop comprises n primary side resistors R_pAnd n primary windings of transformer T, i primary resistance R_piThe voltage regulator is connected with a primary winding of an ith transformer in series to form an ith primary side branch, and n primary side branches are connected in parallel to form A, B two terminals;

the secondary side loop comprises n secondary side resistors R_sAnd secondary windings of the n transformers T, an ith secondary resistor R_siThe voltage regulator is connected with a secondary winding of the ith transformer in series to form an ith secondary branch, and the n secondary branches are connected in parallel to form C, D two terminals.

Preferably, the n transformers T each include an air gap and have the same turns ratio.

Preferably, the primary resistance R is varied_pSecondary side resistance R_sAnd transformingAnd the parameters of the transformer T realize the characteristics of symmetrical fractional order transformers with different orders.

The construction method of the symmetrical fractional order transformer construction circuit comprises the following steps:

s1, determining basic parameters of a symmetrical fractional order transformer corresponding to a construction circuit;

s2, determining the turn ratio of a transformer T in the construction circuit;

s3, determining the number n of primary side resistors and the number R of the primary side resistors in the constructed circuit_piSecondary side resistance R_siAnd the ith transformer T_iExcitation inductance L_miA value of (d);

s4, determining primary side leakage inductance L of n transformers T in structural circuit_pSecondary leakage inductance L_sThe values of the primary winding resistance and the secondary winding resistance.

Preferably, the basic parameters include the order α of the symmetrical fractional order transformer and the inductance L of the primary side inductor₁Inductance value L of secondary side inductor₂；

According to inductance L of primary side inductance₁Inductance value L of secondary side inductor₂Determining the turn number transformation ratio k of n transformers T in the constructed circuit to be:

preferably, based on Oustaloup rational approximation algorithm, the order N of the filter is selected and the frequency range (omega) is selected_b,ω_h) Inductance to primary side inductance of L₁And the fractional order inductance element with the order of alpha is reasonably approximated to obtain an approximate equivalent circuit of the corresponding fractional order inductance, wherein omega is_bIs the lower limit of said frequency band, ω_hIs the upper limit of the frequency band;

preferably, the approximate equivalent circuit of the fractional order inductance is formed by connecting N resistors R in series_iAnd an inductance L_iThe branches are connected in parallel.

Preferably, the number N of the primary resistors is the same as the order N of the filter.

Preferably, the resistance R of the approximate equivalent circuit of the fractional order inductance_iFor constructing a primary resistor R in the circuit_pi(ii) a Inductance L of approximate equivalent circuit of fractional order inductance_iFor constructing the ith transformer T in the circuit_iExcitation inductance L_mi(ii) a Secondary side resistance R_siIs a primary side resistor R_piIs/are as followsAnd (4) doubling.

Preferably, the primary side leakage inductance L_pSet as excitation inductance L_m1% -3% of (A), secondary leakage inductance L_sSet as primary side leakage inductanceThe primary winding resistance and the secondary winding resistance are both set to be less than 10m omega.

The verification steps of the symmetrical fractional order transformer structure circuit are as follows:

and verifying the open port characteristic of the symmetrical fractional order transformer construction circuit through Psim simulation. The port characteristics should satisfy: when the secondary side is open, the impedance of the primary side is expressed as the order of alpha and the inductance of L₁Fractional order inductance of (a); when the primary side is open, the secondary side open circuit impedance is expressed as order alpha and inductance L₂Fractional order inductance of (a);

through Psim simulation verification, under the same load, the fractional order transformer structure circuit and the fractional order transformer equivalent circuit have basically the same characteristics.

Through Psim simulation verification, in the same flyback converter, the fractional order transformer structure circuit and the fractional order transformer equivalent circuit show basically the same characteristics.

Compared with the prior art, the invention has the following beneficial effects:

the symmetrical fractional order transformer construction circuit can truly simulate the characteristics of a symmetrical fractional order transformer and realize the construction of the symmetrical fractional order transformer. The construction of symmetrical fractional order transformers of different orders can be realized by changing the parameters of the primary resistor, the secondary resistor and the transformer. The structure circuit expands the integral-order transformer element into the fractional-order field, and has important significance on circuit theory and actual circuit research.

Drawings

FIG. 1 is a basic block diagram of a symmetrical fractional order transformer configuration circuit of the present invention;

FIG. 2 is a circuit model diagram of a mutual inductance version of the symmetric fractional order transformer of the present invention;

FIG. 3 is an equivalent circuit diagram of the symmetrical fractional order transformer of the present invention;

FIG. 4 is a circuit diagram of 0.95 th and 0.9 th fractional order 1mH inductance approximation circuit under Oustaloup rational approximation

FIG. 5 shows frequency domain waveforms of the open-circuit impedance at the primary side of the circuit constructed by the integer-order transformer and the fractional-order transformers of 0.95 order and 0.9 order;

FIG. 6 shows frequency domain waveforms of secondary side open circuit impedance of the circuit of the integral step transformer, 0.95 step and 0.9 step fractional step transformer;

FIG. 7 shows the primary side voltage and current waveforms of the circuit and its equivalent circuit constructed by an integer-order transformer, 0.95-order and 0.9-order fractional-order transformers under the same resistive load;

FIG. 8 shows the secondary side voltage and current waveforms of the structural circuit of the integer-order transformer, the fractional-order transformers of 0.95 order and 0.9 order and the equivalent circuit thereof under the same resistive load;

FIG. 9 is a graph of the waveforms of the primary current and the output voltage of the structural circuit of the integer-order transformer, the fractional-order transformers of 0.95 order and 0.9 order and the equivalent circuit thereof under the same flyback converter;

fig. 10 is a waveform diagram of the primary side current and the output voltage of the circuit constructed by the integer-order transformer, the 0.95-order and the 0.9-order fractional-order transformers and the equivalent circuit thereof in the steady state under the same flyback converter.

Detailed Description

To further illustrate the content and features of the present invention, the following description of specific embodiments of the present invention is provided in conjunction with the accompanying drawings, but the present invention is not limited thereto.

As shown in fig. 1, the symmetrical fractional order transformer of the present embodiment configures a circuit. The construction circuit comprises a primary side loop and a secondary side loop. The primary side loop comprises n primary side resistors R_pAnd n primary windings of transformer T, i primary resistance R_piThe voltage regulator is connected with a primary winding of an ith transformer in series to form an ith primary side branch, and n primary side branches of a primary side are connected in parallel to form A, B two terminals;

the secondary side loop comprises n secondary side resistors R_sAnd secondary windings of the n transformers T, an ith secondary resistor R_siThe secondary winding of the ith transformer is connected in series to form an ith secondary branch, and the n secondary branches on the secondary side are connected in parallel to form C, D two terminals.

The n transformers T all contain air gaps and have the same turn ratio. And considering practical conditions, the n transformers T have small leakage inductance and winding loss, but the leakage inductance and the winding loss are small.

Fig. 2 shows a circuit model of a symmetric fractional order transformer structure circuit shown in fig. 1, which corresponds to a symmetric fractional order transformer mutual inductance form, and the port characteristics of the circuit model are as follows:

in the formula (I), the compound is shown in the specification,as a fractional order differential operator, u₁Is the primary side voltage u₂Is the secondary side voltage, i₁Is a primary side current i₂Is a secondary side current, L₁Is a primary side inductor, L₂The secondary side inductance is M is mutual inductance, and the order is alpha. Compared with a common fractional order mutual inductance model, the fractional order transformer mutual inductance mode neglects leakage inductance and loss resistance and should have

In this embodiment, a circuit model equivalent circuit in the form of fractional order transformer mutual inductance is shown in fig. 3, and the port characteristics of the equivalent circuit are as follows:

in the formula L_mK is the turn ratio when L is the excitation inductance_m＝L₁，The circuit shown in fig. 3 is completely equivalent to the circuit shown in fig. 2. It should be noted that the circuits shown in fig. 2 and 3 are neglected for leakage inductance and loss resistance, while the transformer in the circuit of fig. 1 has very small loss and leakage inductance, so that both will have errors within an acceptable range.

According to the steps provided by the present invention, the primary side inductors L of 0.95 order and 0.9 order are constructed in this embodiment₁＝1mH/s^1-αSecondary side inductor L₂＝0.25mH/s^1-αA fractional order transformer. Further, in order to make the fractional order characteristic of the structural circuit more obvious, this embodiment also compares with an integer order transformer with a primary side inductance of 1mH and a secondary side inductance of 0.25 mH.

In this embodiment, only a fractional order transformer with the same primary inductance and secondary inductance is constructed. Thus, according to the primary inductance L₁And secondary inductance L₂The turns ratio of the transformer in the 0.95 th and 0.9 th fractional order transformer configuration circuit can be determined

Based on Oustaloup rational approximation algorithm, the order N of the selected filter is 9, and the frequency range is (omega)_b,ω_h)＝(0.01,10⁷)，ω_bIs the lower limit of said frequency band, ω_hFor the upper limit of the frequency band, an inductance value of 1mH/s of 0.95 order can be obtained^1-αAnd an inductance of order 0.9 of 1mH/s^1-αThe fractional order inductance approximation circuit is shown in FIG. 4The corresponding circuit parameters are shown in table 1 and table 2, respectively.

TABLE 1 order alpha of 0.95, inductance of 1mH/s^1-αApproximate circuit parameters of fractional order inductance of

TABLE 2 Induction value of 1mH/s with an order alpha of 0.9^1-αApproximate circuit parameters of fractional order inductance of

The number n of primary resistors in the 0.95-order and 0.9-order fractional order transformer structure circuit can be determined as 9, and the primary resistor R_piSecondary side resistance R_siAnd ith transformer T_iExcitation inductance L_miIt can be determined as: r_pi＝R_i、R_si＝R_i/k²＝R_i/4、L_mi＝L_i。

Furthermore, a transformer T in a 0.95 order and 0.9 order fractional order transformer structure circuit_iPrimary side leakage inductance L_pi＝0.01L_miMinor edge leakage inductance L_si＝0.0025L_miThe primary winding resistance and the secondary winding resistance are both 0.001 omega.

The construction circuit of the fractional order transformer with 0.95 order and 0.9 order is verified:

the secondary side of the circuit constructed by the integer-order transformer and the fractional-order transformer is opened, and sine voltage is added to the primary side of the circuit constructed by the integer-order transformer and the fractional-order transformer, so that the primary side open-circuit impedance frequency domain waveforms of the circuit constructed by the integer-order transformer and the fractional-order transformer with 0.95 order and 0.9 order are shown as the star marks in fig. 5. The solid line in fig. 5 represents the corresponding frequency domain waveform diagram of the fractional order inductive impedance theory.

The frequency domain waveforms of the open-circuit impedance at the secondary side of the circuit constructed by the integer-order transformer and the fractional-order transformer are shown by the asterisk in fig. 6, and the frequency domain waveforms of the open-circuit impedance at the secondary side of the circuit constructed by the integer-order transformer and the fractional-order transformer with 0.95 order and 0.9 order are obtained by adding sinusoidal voltage to the secondary side. The solid line in fig. 6 represents the corresponding frequency domain waveform diagram of the fractional order inductive impedance theory.

As can be seen from fig. 5 and 6, the open circuit characteristics of the 0.95 th and 0.9 th fractional order transformer structure circuits satisfy: secondary side open circuit, primary side open circuit impedance is shown as order alpha, inductance value is L₁Fractional order inductance of (a); open circuit at primary side and open circuit at secondary side with order of alpha and inductance of L₂Fractional order inductance of (a);

sine waves with amplitude of 220V and frequency of 20kHz are added to primary sides of construction circuits of an integer-order transformer, a 0.95-order fractional-order transformer and an equivalent circuit of the construction circuits, and secondary sides are connected in series with resistance R_LA load resistance of 10 Ω. The voltage and current on the primary side of the integer transformer, the fractional transformer structure circuits of 0.95 order and 0.9 order and their equivalent circuits are shown in fig. 7, and the voltage and current on the secondary side are shown in fig. 8. According to fig. 7 and 8, it can be seen that when the loads are the same, the characteristics of the fractional order transformer construction circuit and the equivalent circuit are basically the same.

The integral transformer, the 0.95-order and 0.9-order fractional transformer construction circuits and equivalent circuits thereof are all applied to the same flyback converter. The parameters of the flyback converter are selected as an input voltage source U_inThe switching frequency F is 20kHz, the duty ratio D is 0.5, the output capacitance C is 100 uf, and the output resistance R is 10 Ω, 20V. Fig. 9 shows input side currents and output side capacitor voltages of the integer-order transformer, 0.95-order and 0.9-order fractional-order transformer structural circuits and equivalent circuits thereof, and fig. 10 shows waveforms of the input side currents and the output side capacitor voltages in steady states. It can be seen from fig. 9 and 10 that in the same flyback converter, the characteristics exhibited by the fractional transformer structure circuit and the equivalent circuit are substantially the same, and the error is mainly caused by the fact that the transformer T in the structure circuit takes leakage inductance and winding resistance into consideration。

From the above analysis, under the conditions of no-load, resistive load and the same flyback converter, the characteristic presented by the symmetrical fractional order transformer structure circuit of the invention is basically consistent with the equivalent circuit of the symmetrical fractional order transformer. This shows that the circuit of the present invention can truly simulate a symmetric fractional order transformer. Therefore, the circuit of the invention is correct and feasible and is worth popularizing.

The above embodiments are preferred embodiments of the present invention, but the present invention is not limited to the above embodiments, and any other changes, modifications, substitutions, combinations, and simplifications which do not depart from the spirit and principle of the present invention should be construed as equivalents thereof, and all such changes, modifications, substitutions, combinations, and simplifications are intended to be included in the scope of the present invention.